# Microcanonical ensemble for system of harmonic oscillators

1. Apr 1, 2010

### everyday847

1. The problem statement, all variables and given/known data

A system consists of 3N (N >> 1) independent, identical, but distinguishable one-dimensional oscillators. This is relevant in that the atoms in a solid are sitting around their equilibrium positions. Assume that every atom constitutes an independent oscillator and all oscillators are characterized by the angular frequency ω. From quantum mechanics, the allowed energies of a 1D oscillator with angular frequency ω is given by

$$\epsilon = \hbar \omega/2 + 3\hbar \omega/2 + ... + (n +1/2) \hbar \omega + ...$$

where \hbar \omega / 2 is the ground state energy of the oscillator. For the 3N-oscillator system, given that the total energy is given as follows

$$E = M\hbar \omega + (3/2) N\hbar \omega$$

where M is an integer that describes the extent of excitations in the system and
can be taken to be M >> 1 .

a) Find density of states, Ω(M, N) corresponding to the total energy
in Eq. (2).
b) Using microcanonical ensemble, show that the internal energy
can be written as follows.

$$\bar{E} = 3N\left(\frac{\hbar \omega}{2} + \frac{\hbar \omega}{e^{\hbar \omega / kT}-1} \right)$$

2. Relevant equations

Clearly the above definitions are relevant. Also relevant ought to be the fundamental equation of the microcanonical ensemble: that the energy derivative of \ln \Omega is equal to beta.

3. The attempt at a solution

Well, there are M excitations distributed among 3N degrees of freedom. But I can't really discuss the number of ways they can be distributed (thus enumerating the number of states) within the microcanonical ensemble: one needs the grand canonical ensemble because the number of excitations is fixed.

I can certainly handle the second part using a canonical ensemble because it's straightforward to take the CPF of a harmonic oscillator, take the 3N power, and pump. Microcanonical, not so much. Actually enumerating the states? Certainly not sure I can do that.

I'm actually fairly muddled here. Can I just assume that \Omega is a Gaussian? If so, can I simply then approximate that, since beta is approximately 3N/\bar{E}, \bar{E} is approximately 3N divided by the energy derivative of \ln \Omega? That all seems wrong.

Last edited: Apr 1, 2010